0. Introduction: In the last twenty years, several remarkable
results have been established in the context of infinite dimensional
Banach space theory. Most of these results emphasize the interplay
between the topological, geometrical and measure theoretical
structures of a Banach space. Here are two well known prototypes of
such interrelations. They are due to the combined efforts of several
authors and we refer the reader to the books [8] and [14] for a
detailed account of their history and for the notions involved in
their statements.
Theorem (A): Let X be a Banach space and let D be a closed convex
bounded subset of X. The following properties are then equivalent:
(i) All D-valued martingales norm converge almost surely,
(ii) Every non-empty subset of D has slices of arbitrarily small
diameter.
Theorem (B): Let Y be a separable Banach space and let D be a
* *
w -compact convex subset of Y The following properties are then
equivalent:
(i) All D-valued martingales converge in the Pettis-norm.
**
(ii) Every functional in Y is the pointwise limit on D of a
bounded sequence of elements in Y.
A set verifying the conditions of Theorem (A) is then said to
have the Radon-Nikodym property (R.N.P) while a set verifying those of
Theorem (B) is said to have the Weak Radon-Nikodym property
(W.R.N.P). In both cases, the set is then the norm closed convex hull
of its extreme points.
The concept of a Radon-Nikodym set turned out to be central in
the study of extremal structures in convex sets, integral
representations and problems involving non-linear optimization ([8],
[14], [19]). On the other hand, the weak Radon-Nikodym property (for
w*-compact sets) is closely related to the classical Baire theory of
functions and its relatively recent resurgence with the deep theorems
of Bourgain-Fremlin-Talagrand (B.F.T [5]) following the pioneering
work of Rosenthal [38] and Odell-Rosenthal [33].
However, even though these two concepts are now well understood
Received by the editors September 26, 1986.
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